(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(app(comp, f), g), x) → app(f, app(g, x))
app(twice, f) → app(app(comp, f), f)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(app(comp, f), g), x) → APP(f, app(g, x))
APP(app(app(comp, f), g), x) → APP(g, x)
APP(twice, f) → APP(app(comp, f), f)
APP(twice, f) → APP(comp, f)
The TRS R consists of the following rules:
app(app(app(comp, f), g), x) → app(f, app(g, x))
app(twice, f) → app(app(comp, f), f)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(app(comp, f), g), x) → APP(g, x)
APP(app(app(comp, f), g), x) → APP(f, app(g, x))
The TRS R consists of the following rules:
app(app(app(comp, f), g), x) → app(f, app(g, x))
app(twice, f) → app(app(comp, f), f)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
comp = comp
app(x1, x2) = app(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- APP(app(app(comp, f), g), x) → APP(g, x) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- APP(app(app(comp, f), g), x) → APP(f, app(g, x)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE